Mrs. Clinton’s resolve to attend the event drew admiration from some who know her. “It speaks to the depth of her character, her patriotism and her broader faith that often got lost in the campaign and was so often undernoticed,” said Christine C. Quinn, a former New York City Council speaker and mayoral candidate in town for the Women’s March on Saturday.

On Friday morning, Mrs. Clinton wrote on Twitter: “I’m here today to honor our democracy & its enduring values. I will never stop believing in our country & its values.”

Mr. Trump and Mrs. Clinton did not shake hands at the swearing-in, but after the inauguration, the Clintons attended a luncheon in the Capitol, where the new president said he was “very, very honored” that the former first couple was there and asked them to stand for a round of applause. The Clintons, who sat at a well-positioned table with Mr. Trump’s daughter Tiffany, and others close to the new president, rose from their seats with tight grins.

“There’s nothing more I can say because I have a lot of respect for those two people, so thank you all for being here,” Mr. Trump said.

Aside from the occasional appearance at a Broadway show or walk in the woods around her home in Chappaqua, N.Y., Mrs. Clinton has kept a low profile since her election loss in November. As her husband and daughter, Chelsea Clinton, dive into their efforts at the Clinton Foundation, Mrs. Clinton has given no signal about what she will do next. “She’s just not there yet,” said a friend who talked to her recently.